Is -12.34545454… rational or irrational 

The number \(-12.34545454\ldots\) is **rational**. 

This is because the decimal is repeating (the sequence "54" repeats indefinitely), and a number with a repeating decimal is always rational. A rational number can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).

If we let \(x = -12.34545454\ldots\), we can express this number as a fraction by using the following steps:

1. Let \(x = -12.34545454\ldots\)
2. Multiply by 1000 (since the repeating part has 2 digits): \(1000x = -12345.454545\ldots\)
3. Subtract the original equation from this: \(1000x - x = -12345.454545\ldots + 12.34545454\ldots\)
4. Simplifying: \(999x = -12333.10909\ldots\)
5. The repeating part cancels out, leaving \(x = \frac{-12333.10909\ldots}{999}\).

Thus, \(x\) can be expressed as a fraction, confirming that the number is rational.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you convert a repeating decimal to a fraction in general?
2. What is the difference between rational and irrational numbers?
3. Can all repeating decimals be classified as rational numbers?
4. How would you convert \( -12.34545454\ldots \) to its exact fractional form?
5. What are some examples of irrational numbers?

**Tip:** A number with a non-repeating, non-terminating decimal is always irrational.

Explore why -12.34545454… is a rational number with a repeating decimal sequence. Learn how to convert repeating decimals to fractions with step-by-step explanations and examples. Suitable for Grades 6-8 students and anyone interested in understanding rational numbers.

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